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Metamath Proof Explorer

Theorem dihglblem2aN

Description: Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihglblem.b 
|- B = ( Base ` K )
dihglblem.l 
|- .<_ = ( le ` K )
dihglblem.m 
|- ./\ = ( meet ` K )
dihglblem.g 
|- G = ( glb ` K )
dihglblem.h 
|- H = ( LHyp ` K )
dihglblem.t 
|- T = { u e. B | E. v e. S u = ( v ./\ W ) }
Assertion dihglblem2aN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )

Proof

Step Hyp Ref Expression
1 dihglblem.b 
 |-  B = ( Base ` K )
2 dihglblem.l 
 |-  .<_ = ( le ` K )
3 dihglblem.m 
 |-  ./\ = ( meet ` K )
4 dihglblem.g 
 |-  G = ( glb ` K )
5 dihglblem.h 
 |-  H = ( LHyp ` K )
6 dihglblem.t 
 |-  T = { u e. B | E. v e. S u = ( v ./\ W ) }
7 6 a1i 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T = { u e. B | E. v e. S u = ( v ./\ W ) } )
8 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> S =/= (/) )
9 n0 
 |-  ( S =/= (/) <-> E. z z e. S )
10 8 9 sylib 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> E. z z e. S )
11 hllat 
 |-  ( K e. HL -> K e. Lat )
12 11 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> K e. Lat )
13 simplrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> S C_ B )
14 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> z e. S )
15 13 14 sseldd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> z e. B )
16 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
17 16 ad3antlr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> W e. B )
18 1 3 latmcl 
 |-  ( ( K e. Lat /\ z e. B /\ W e. B ) -> ( z ./\ W ) e. B )
19 12 15 17 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> ( z ./\ W ) e. B )
20 eqidd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> ( z ./\ W ) = ( z ./\ W ) )
21 oveq1 
 |-  ( v = z -> ( v ./\ W ) = ( z ./\ W ) )
22 21 rspceeqv 
 |-  ( ( z e. S /\ ( z ./\ W ) = ( z ./\ W ) ) -> E. v e. S ( z ./\ W ) = ( v ./\ W ) )
23 14 20 22 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> E. v e. S ( z ./\ W ) = ( v ./\ W ) )
24 ovex 
 |-  ( z ./\ W ) e. _V
25 eleq1 
 |-  ( w = ( z ./\ W ) -> ( w e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( z ./\ W ) e. { u e. B | E. v e. S u = ( v ./\ W ) } ) )
26 eqeq1 
 |-  ( u = ( z ./\ W ) -> ( u = ( v ./\ W ) <-> ( z ./\ W ) = ( v ./\ W ) ) )
27 26 rexbidv 
 |-  ( u = ( z ./\ W ) -> ( E. v e. S u = ( v ./\ W ) <-> E. v e. S ( z ./\ W ) = ( v ./\ W ) ) )
28 27 elrab 
 |-  ( ( z ./\ W ) e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) )
29 25 28 bitrdi 
 |-  ( w = ( z ./\ W ) -> ( w e. { u e. B | E. v e. S u = ( v ./\ W ) } <-> ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) ) )
30 24 29 spcev 
 |-  ( ( ( z ./\ W ) e. B /\ E. v e. S ( z ./\ W ) = ( v ./\ W ) ) -> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
31 19 23 30 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
32 n0 
 |-  ( { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) <-> E. w w e. { u e. B | E. v e. S u = ( v ./\ W ) } )
33 31 32 sylibr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ z e. S ) -> { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) )
34 10 33 exlimddv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> { u e. B | E. v e. S u = ( v ./\ W ) } =/= (/) )
35 7 34 eqnetrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )